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I would like to ask why are linear motors not being considered for mainstream use on main rail lines. If there was a linear motor system then you would not need to maintain vast amounts of cabling above electrified railway lines.
A major improvement would be that linear motors could be switch on only when a train passes over the motor, which is a big energy saving to take into consideration in relation to overhead cables which need a constant energy supply.
The could be cost saving over constructing and maintaining overhead cabling.
In terms of health and safety the linear motors look to be a safer option than overhead power cables because they would be switched of until a train passes over them unlike overhead power cables that are on all the time.

Why is there not any implementation of linear motor systems on main lines? Is there something that I have overlooked in my argument?

In principal, yes, it's a fantastic idea, but in practice, it would mean major changes to all UK inferstructure, and rolling stock.

The 25kV OHL system can supply power to traction motors at any point, and for any vehicle, be it a locomotive, or multiple unit.

Safety wise, I havn't seen any safety issues in a long time with OHL and people killing or injuring themselves, short of idiots trying to steal cable.

Cost wise, how much would these linear motors cost to install, bearing in mind they'd proberbly require copper windings that would get stolen very frequently on the UK network, as anyone who commutes through Newton Heath or Clifton will know. You'd also need these at every signal point and long enough to bring the units back up to linespeed again, and for some units and trains, this can be all the way to the next signal block, so realistically to maintain performance you'd need to install these linear motors over the whole track, we're now putting windings everywhere, and this is a significant amount more cabling than OHL uses.

Then there is the matter of the fact that OHL doesn't just feed traction motors, it also provides the ETH supply (A legacy term Electric Train Heating) that provides all ancillary services, such as lighting, air conditioning, toilets, heating, plug sockets, tilt motors (On 221 and 390 units) control systems etc.

Control systems brings be onto another thing, this would introduce an additional point of failiure, where currently if a traction motor winding fails on say, a Class 380 unit, this motor can be isolated and the unit continue to some sidings where it is out of the way, if the linear motors fail, the line fails and you need a diesel locomotive to haul it away.

Theres also the question of how one would control these from the unit...

Maglev does use this system and it does work, but it would be prohivitavely expensive (and pointless) to fit this to the existing UK system, that is also ubiquitious around Europe for rail systems (Except our loading gauge is smaller at W6, W8 or W10/12 rather than the EU Standard of UIC-GC, that is being employed on all new lines wherever practical in the UK) The 'Standard Gauge' Railway will be here to stay for a while I'm afraid.